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!set gl_author=Sophie, Lemaire
!set gl_keywords=continuous_probability_distribution
!set gl_title=
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<div class="wims_defn"><h4>Definition</h4>
Let \(n) be a positive integer.
The <span class="wims_emph">chi-squared distribution with \(n) degrees
of freedom</span> (denoted by \(\chi^2(n))), is the distribution
of the sum of \(n) independent \(\mathcal{N}(0,1)) distributed random
variables.
Its density function is
<div class="wimscenter">
\( \displaystyle{ x \mapsto \frac{x^{\frac{n}{2}-1}e^{-\frac{x}{2}}}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}1_{x>0 }})
</div>
</div>
<table class="wimsborder wimscenter">
<tr><th>Esprance</th><th>Variance</th><th>Characteristic function</th></tr>
<tr><td>\(n)</td><td>\(2n)</td><td>\((1-2i t)^(-n/2))</td></tr></table>
